We investigate the non-Gaussianity of second-order matter density perturbations induced by primordial gravitational waves (GWs). These tensor induced scalar modes arise from local fluctuations in the GW energy density, which is quadratic in tensor perturbations. The resulting second-order density contrast follows a $\chi$-squared distribution, naturally exhibiting significant non-Gaussianity. We compute the bispectrum of these tensor induced scalar modes and analyze its dependence on various primordial GW power spectra, including scale-invariant, blue-tilted, Gaussian-bump, and monochromatic sources. We find that the bispectrum shape is inherently sensitive to the underlying GW spectrum by construction. In particular, Gaussian-bump and monochromatic sources produce a strong signal peaking in the equilateral configuration that is similar to the effect of scalar induced tensor modes. Our findings reveal a new way to probe primordial GWs via galaxy surveys and highlight a unique feature of tensor induced density perturbations, otherwise mimicking linear ones on subhorizon scales.

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• Cosmology
• Dark energy
• Evolution of the Universe
• Gravitational waves
• Inflation
• Large scale structure of the Universe